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Telescopic Homotopy Theory and Algebraic K-theory

Building on visionary work of Waldhausen, Thomason and Goodwillie, trace methods and chromatic homotopy theory have together illuminated a certain sector of algebraic K-theory. Recently, information has begun to flow in the reverse direction. Redshifted K-theory is presently the only known way of distinguishing between the T(n) and K(n) -local modalities. This workshop aims to bring leading experts in chromatic homotopy theory and algebraic K-theory together in exploring this thread.

Invited speakers: Sanjana Agarwal (UI Bloomington), Benjamin Antieau (Northwestern), Agnes Beaudry (Colorado Boulder), Mark Behrens (Notre Dame), Shay Ben-Moshe (MPI), Shachar Carmeli (Weizmann), Alexander Efimov (Steklov), Gijs Heuts (Utrecht), Mike Hopkins (Harvard), Achim Krause (Oslo), Lennart Meier (Utrecht), Thomas Nikolaus (Münster), Niko Naumann (Regensburg), Arpon Raksit (Northwestern), Maxime Ramzi (Münster), John Rognes (Oslo), Andrew Senger (Maryland), Vladimir Sosnilo (RIKEN), Lior Yanovski (Melbourne), Allen Yuan (Northwestern)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor. 

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the conference and workshops will be sent by CMI or a workshop organizer. 

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Venue: Mathematical Institute, University of Oxford

The Birch and Swinnerton-Dyer Conjecture and Related Problems: Recent Results

Birch and Swinnerton-Dyer proposed their remarkable conjecture linking the arithmetic of an elliptic curve with analytic properties of its L-function over 60 years ago. It has been just over 25 years since the BSD conjecture was listed as one of the Clay Mathematics Institute’s Millennium Prize Problems. This workshop will focus on some of the recent results towards the conjecture and related problems, including generalizations to abelian varieties, questions about ranks, computational work, results over function fields, and arithmetic statistics.

Invited speakers: Manjul Bhargava (Princeton), Ashay Burungale (UT Austin), Ana Caraiani (Imperial), John Cremona (Warwick), Henri Darmon (McGill), Noam Elkies (Harvard), Alex Smith (Northwestern), Naomi Sweeting (Princeton), Ye Tian (CAS), Ziquan Yang (CUHK), Shou-Wu Zhang (Princeton), David Zywina (Cornell)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor. 

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the conference and workshops will be sent by CMI or a workshop organizer. 

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Venue: Mathematical Institute, University of Oxford

Multi-valued Harmonicity

Harmonic functions which are not single-valued arise in classical complex function theory, but these objects have emerged in several different places in more recent research.  Already 50 years ago, Almgren studied the calculus of multi-valued functions in general dimensions as a means to parametrize area-minimizing submanifolds of codimension greater than one, and modern regularity theory in that subject employs this framework as a basic tool. 

More recently, multi-valued harmonic 1-forms and spinors have appeared as limiting blowup profiles for certain new gauge theories with nonimpact gauge group, and in addition when characterising degenerations of calibrated submanifolds in Riemannian manifolds with special holonomy.

More recently still, multi-valued harmonicity arises in the context of thin obstacle problems. The new (and perhaps central) features of these objects are their codimension two branching sets, which are the root of many analytic subtleties. There are now many researchers encountering these objects and proving interesting theorems about multi-valued harmonic objects.

This workshop aims to collect experts from these various branches of geometric analysis to illuminate problems and methods that should be useful to the whole community, and to suggest new directions in the field. 

Speakers: Cristiana De Filippis (Parma) , Federico Franceschini (Stanford), Andriy Haydys (ULB), Siqi He (CAS), Yang Li (Cambridge), Greg Parker (Berkeley) tbc, Alessandra Pluda (Pisa), Laura Schaposnik (Chicago), Anna Skorobogatova (ETH), Ryosuke Takahashi (Tohoku) tbc, Thomas Walpuski (Humboldt), Neshan Wickramasekera (Cambridge), Therese Wu (Houston), Dashen Yan (Stanford)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor. 

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the conference and workshops will be sent by CMI or a workshop organizer. 

Details

Venue: Mathematical Institute, University of Oxford

Hodge Theory and Algebraic Cycles

The Hodge conjecture proposes a mysterious connection between analysis and algebraic geometry. On the 25th anniversary of the Clay Millennium Problems, this workshop will bring together experts from many areas of Hodge theory and algebraic cycles. Some dramatic recent advances include applications of hyperkahler geometry to the Hodge conjecture, and a better understanding of special subvarieties in Hodge theory and number theory.

Speakers: Ekaterina Amerik (Paris-Saclay and HSE), Benjamin Bakker (UIC), Anna Cadoret (Jussieu), Francois Charles (ENS), Daniel Huybrechts (Bonn), Bruno Klingler (Humboldt), Eyal Markman (UMass), Alexander Petrov (MIT), Stefan Schreieder (Hannover), Junliang Shen (Yale), Salim Tayou (Dartmouth), Claire Voisin (Jussieu)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

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Venue: Mathematical Institute, University of Oxford

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Zeta and L-functions

In recent years there has been great progress in understanding the distribution of zeros and values of zeta and L-functions.  The workshop will feature talks on some key recent advances, including (i) new zero density estimates for the Riemann zeta-function, (ii) progress on the sub-convexity problem for L-functions, (iii) connections with probability theory and random matrix theory, (iv) progress on understanding moments, both in the number field and function field setting (with emerging connections with problems in algebraic topology), and (v) progress on understanding character sums and multiplicative functions in general.

Speakers: Louis-Pierre Arguin (Oxford), Vorrapan Chandee (KSU), Alexandra Florea (UC Irvine), Adam Harper (Warwick), Kaisa Matomäki (Turku), James Maynard (Oxford), Paul Nelson (Aarhus), Dan Petersen (Stockholm), Ian Petrow (UCL), Lillian Pierce (Duke), Brad Rodgers (Queens)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

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Venue: Mathematical Institute, University of Oxford

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PDE and Fluids

Important mathematical questions regarding the fundamental equations of fluids and plasmas remain unanswered. This workshop will highlight recent advances in the understanding of the Euler equations, the Navier–Stokes equations, the Boltzmann equation, and other fluid and plasma models.

Speakers: Jacob Bedrossian (UCLA), Maria Colombo (EPFL), Diego Córdoba Gazolaz (ICMAT), Tarek Elgindi (Duke), Javier Gómez-Serrano (Brown), Zaher Hani (Michigan), Thomas Hou (Caltech), Alexander Kiselev (Duke), Matthew Novack (Purdue), Laure Saint-Raymond (IHES), Steve Shkoller (UC Davis), Klaus Widmayer (Zürich)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

Image: Leif Ristroph and Jun Zhang, NYU’s Applied Math Lab

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Venue: Mathematical Institute, University of Oxford

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P vs NP and Complexity Lower Bounds

The P vs NP problem remains one of the deepest and most consequential open questions in mathematics and theoretical computer science. To mark the 25th anniversary of the Millennium Prize Problems, this workshop will bring together researchers to reflect on progress, challenges, and evolving perspectives in computational complexity. The event will explore major breakthroughs in complexity theory, highlighting circuit lower bounds, meta-complexity, algebraic and geometric complexity theory, hardness of approximation, proof complexity, and beyond. With a focus on both technical advancements and broader insights into the nature of mathematical difficulty, the workshop will provide a forum for cutting-edge research and lively discussion on the future directions of one of the most celebrated problems in mathematics.

Speakers: Eric Allender (Rutgers), Shuichi Hirahara (NII), Christian Ikenmeyer (Warwick), Valentine Kabanets (Simon Fraser), Michal Koucky (Charles), Nutan Limaye (ITU Copenhagen), Ryan O’Donnell (Carnegie Mellon), Igor Oliveira (Warwick), Toni Pitassi (Toronto), Pavel Pudlák (CAS), Benjamin Rossman (Duke), Shubhangi Saraf (Toronto), Iddo Tzameret (Imperial), Leslie Valiant (Harvard), Ryan Williams (MIT), Henry Yuen (Columbia), David Zuckerman (UT Austin)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor. Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

Details

Venue: Mathematical Institute, University of Oxford

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Patterns in Cohomology of Moduli Spaces

This workshop will explore recent developments in the cohomology of many examples of moduli spaces, such as the classical moduli space of Riemann surfaces, moduli spaces of graphs or smooth manifolds, or related objects such as arithmetic groups. Talks by leading researchers will provide different perspectives on these spaces, and will explain new connections to homotopy theory and representation theory, new patterns in cohomology, and applications to arithmetic and geometry.

Speakers:  Alexander Berglund (Stockholm), Carel Faber (Utrecht) tbc, Benson Farb (Chicago), Ezra Getzler (Northwestern), Paul Gunnells (U Mass), Richard Hain (Duke), Alexander Kupers (Harvard), Eduard Looijenga (Utrecht and Tsinghua), Sam Payne (Austin), Dan Petersen (Stockholm), Andrew Putman (Notre Dame), Steven Sam (UCSD), Orsola Tommasi (Padova), Karen Vogtmann (Warwick), Craig Westerland (Minnesota)

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Venue: Mathematical Institute, University of Oxford

Water Waves and Related Fluid Models

The aim of this workshop is to discuss recent developments in the theory of water waves and related fluid models.  The main topics to be discussed are: local and global regularity, dynamical formation of singularities, and numerical methods.

In the past five years, new methods have emerged in the study of global solutions of quasilinear evolutions, inspired by the advances in semilinear theory.  The basic idea is to combine the classical energy and vector-fields methods with a new ingredient, namely refined analysis of the Duhamel formula using the Fourier transform method.  These new methods led to major progress in understanding the global dynamics of small-data solutions of many physically relevent Quasilinear models in 3 and 2 dimensions, including the construction of the first nontrivial global solutions in several classical water wave models.  At the same time, new methods have been developed to study the dynamics of large-data solutions, and the formation of certain types of interface singularities, such as the so-called splash singularity, wherein a locally smooth interface self-intersects in finite time.  The workshop will survey these recent developments and future research directions.

Speakers: Thomas Alazard (ENS), Peter Constantin (Princeton), Diego Cordoba (ICMAT), Daniel Coutand (Heriot Watt), Charles Fefferman (Princeton), Javier Gomez Serrano (Princeton), Alex Ionescu (Princeton), Alex Kiselev (Rice), David Lannes (Bordeaux), Victor Lie (Purdue), Nader Masmoudi (Courant), Benoit Pausader (Princeton), Fabio Pusateri (Princeton), Gregory Seregin (Oxford), Jalal Shatah (Courant), Steve Shkoller (UC Davis), Chongchun Zeng (Georgia Tech)

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Venue: Mathematical Institute, University of Oxford

New Frontiers in Probabilistic and Extremal Combinatorics

Speakers: Matija Bucic (Princeton), David Conlon (Caltech), Tim Gowers (Cambridge), Ben Green (Oxford), Annika Heckel (Uppsala), Oliver Janzer (Cambridge), Jeff Kahn (Rutgers), Peter Keevash (Oxford), Matt Kwan (IST), Imre Leader (Cambridge), Shoham Letzter (UCL), Dor Minzer (MIT), Richard Montgomery (Warwick), Will Perkins (GaTech), Alexey Pokrovskiy (UCL), Oliver Riordan (Oxford), Wojciech Samotij (Tel Aviv), Lisa Sauermann (Bonn), Mehtaab Sawhney (MIT), Asaf Shapira (Tel Aviv), Jacques Verstraete (UCSD)

Registration:  Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

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Venue: Mathematical Institute, University of Oxford

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The P=W Conjecture in Non Abelian Hodge Theory

The P=W Conjecture in the Non Abelian Hodge Theory of compact Riemann surfaces and for the general complex linear group was formulated by M.A. de Cataldo, T. Hausel and L. Migliorini in 2008. It was proved in the special cases of rank 2 and subsequently of genus 2. It has been proved in all rank and genera in 2022 by two independent teams. The conjecture is still open for other complex reductive groups and for higher dimensional complex projective manifolds. Since its inception, the conjecture has inspired and motivated work in several related areas: Non Abelian Hodge Theory (also in positive and mixed characteristics) such as Hitchin systems and character varieties, mirror symmetry, hyperkahler geometry, geometric representation theory, Donaldson-Thomas invariants, cluster varieties.

This workshop aims at bringing together experts in these related areas with the purpose of bringing into focus the state of the art in the subject, sharpening the vision for the near future and laying the ground for new collaborations. 

Speakers: Ben Davison (Edinburgh), Camilla Felisetti (Modena and Reggio Emilia), Andres Fernandez Herrero (Penn), Andrew Harder (Lehigh), Victoria Hoskins (Radboud), Tasuki Kinjo (Kyoto), Emmanuel Letellier (Jussieu), Anton Mellit (Vienna), Tony Pantev (Penn), Fernando Rodriguez Villegas (ICTP), Olivier Schiffmann (Paris-Saclay), Junliang Shen (Yale), Olga Trapeznikova (IST), Qizheng Yin (Peking), Siqing Zhang (IAS), Zili Zhang (Tongji)

Registration: Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organiser. 

Details

Venue: Mathematical Institute, University of Oxford

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New Advances in the Langlands Program: Geometry and Arithmetic

The last decade has seen significant progress in our understanding of the arithmetic of the Langlands program over number fields. Some of the inter-related themes to be explored in this workshop include: the proof of new cases of Langlands reciprocity, linking Galois representations and automorphic forms; breakthroughs in understanding the geometry of Shimura varieties (particularly their analytic geometry over p-adic fields); the growing role of moduli spaces of Galois representations.

Speakers: George Boxer (Imperial), Matthew Emerton (Chicago), Tony Feng (Berkeley), Jessica Fintzen (Bonn), Toby Gee (Imperial), Ian Gleason (MPIM), Teruhisa Koshikawa (Kyoto), Si Ying Lee (Stanford), Jeffrey Manning (Imperial), Alice Pozzi (Bristol), Juan Esteban Rodriguez Camargo (Columbia), Peter Scholze (Bonn), Chris Skinner (Princeton), Naomi Sweeting (Harvard), Jack Thorne (Cambridge), Pol van Hoften (VU Amsterdam), Andrew Wiles (Oxford), Sarah Zerbes (ETH), Mingjia Zhang (Princeton)

This workshop is also supported by the ERC Starting Grant “P-adic arithmetic geometry, torsion classes, and modularity” (grant agreement No. 804176).

Registration: Participation in the workshop is by invitation, but a limited number of additional places is available.  To register for the Clay Research Conference or to register interest in a workshop, please email Naomi Kraker, providing the name of your institution and stating which workshop you wish to attend. Students please also provide a letter of reference from your supervisor.  Registration is now closed.

HOTEL SCAM ALERT:  Some speakers may be contacted via email or telephone by Travellerpoint.org or similar.  Please be advised that we use no external organizing service.  All emails regarding the workshop will be sent by CMI or a workshop organizer.

Details

Venue: Mathematical Institute, University of Oxford

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Schedule Abstracts